Problem 24 - Algebra and Trigonometry

Question

\(13-24\) . Find the degree measure of the angle with the given radian measure. $$ -\frac{13 \pi}{12} $$

Step-by-Step Solution

Answer

The degree measure is \(-195^\circ\).

Step 1: Understand the Relation Between Radians and Degrees

To convert an angle from radians to degrees, you can use the conversion factor, which is \(180^\circ = \pi \text{ radians}\). This means to convert radians to degrees, you multiply by \(\frac{180}{\pi}\).

Step 2: Set Up the Conversion

Given the radian measure \(-\frac{13\pi}{12}\), we use the conversion factor \(\frac{180}{\pi}\) to set up the conversion to degrees. The formula is:\[-\frac{13\pi}{12} \times \frac{180}{\pi}\].

Step 3: Cancel Out the \(\pi\) Terms

In the multiplication, the \(\pi\) in the numerator and denominator will cancel each other out, simplifying the expression to:\[-\frac{13 \times 180}{12}\].

Step 4: Perform the Multiplication and Division

Now, calculate the multiplication and division:First calculate the multiplication part:\(13 \times 180 = 2340\).Then divide by 12:\[ \frac{2340}{12} = 195\].

Step 5: Apply the Negative Sign

Remember that the original expression was negative, so the final degree measure will also be negative:\(-195^\circ\).

Key Concepts

Angle Measurement

When learning about angles, understanding how they are measured is important. Two common units for measuring angles are radians and degrees.
Radians are often used in trigonometry and calculus, especially because of how nicely they correspond with quantities like \(\pi\) and the unit circle.
On the other hand, degrees are more intuitive for everyday situations, like when talking about angles in geometry or design. Knowing how to switch between these measurements lets you talk about angles no matter the context you're working in.

• Radians use \(\pi\) to express measurements.
• Degrees split a full circle into 360 equal parts.

Understanding these systems means being versatile in different mathematical contexts.

Degree Measure

Degrees are the most common way to measure angles. We often see them in everyday life.
A complete circle is divided into 360 degrees, making it easy to work with because you can split circles and angles into equal parts more simply than other units.
The degree measure is usually denoted by the symbol \(^\circ\). Here are some key facts about degrees:

• 1 degree is \(\frac{1}{360}\) of a full circle.
• Because of its convenience, many instruments use degrees to measure angles.
• Often used in navigation, design, and construction.

When you're converting from radians to degrees, it's important to remember how these 360 parts relate to the angle you're interested in. This makes handling angles in degrees very practical for various applications.

Conversion Factor

Conversions between different units of measurement often rely on a specific conversion factor.
For angle conversions, such a factor helps you change radians to degrees or vice versa.
The conversion factor can simplify the process and ensure consistency in calculations. For converting radians to degrees, you use the fact that \(180^\circ = \pi\) radians. This gives us the conversion factor \(\frac{180}{\pi}\).

• This factor means every \(\pi\) radians equals 180 degrees.
• To convert radians to degrees, you multiply by \(\frac{180}{\pi}\).
• To convert back (degrees to radians), multiply by \(\frac{\pi}{180}\).

This conversion factor maintains the relationship between these measurements and makes it easy to switch from radians to a system you're more familiar with, like degrees.